Note that if \(\mathcal {S}\)  is a free semigroup and \(S_i\) -fibred IFSs satisfy the OSC with respect to \((0,1)\)  for all \(i\in \mathcal {I}\) , then \(E_\eta \)  is the attractor of a non-autonomous self-similar IFS, as introduced and studied in [1]}.
In this situation, it holds that
\(\sup _{\eta \in \Omega }E_\eta =\max \lbrace a_i\rbrace _{i\in \mathcal {I}}\) 
